On the convergence rates of generalized conditional gradient method for fully discretized Mean Field Games

TL;DR

This paper analyzes the convergence rates of the generalized conditional gradient method applied to fully discretized Mean Field Games, providing explicit error estimates that account for discretization and iteration errors.

Contribution

It offers the first rigorous analysis combining discretization and iteration errors for GCG in fully discretized MFG systems, with explicit convergence rate estimates.

Findings

Derived explicit error bounds depending on mesh sizes and iterations

Established discrete maximum principles under structural assumptions

Numerical experiments confirm theoretical convergence rates

Abstract

We study convergence rates of the generalized conditional gradient (GCG) method applied to fully discretized Mean Field Games (MFG) systems. While explicit convergence rates of the GCG method have been established at the continuous PDE level, a rigorous analysis that simultaneously accounts for time-space discretization and iteration errors has been missing. In this work, we discretize the MFG system using finite difference method and analyze the resulting fully discrete GCG scheme. Under suitable structural assumptions on the Hamiltonian and coupling terms, we establish discrete maximum principles and derive explicit error estimates that quantify both discretization errors and iteration errors within a unified framework. Our estimates show how the convergence rates depend on the mesh sizes and the iteration number, and they reveal a non-uniform behavior with respect to the iteration. Moreover, we prove that higher convergence rates can be achieved under additional regularity assumptions on the solution. Numerical experiments are presented to illustrate the theoretical results and to confirm the predicted convergence behavior.